A current I=10A flows in a ring of radius `r_0=15 cm` made of a very thin wire. The tensile strength of the wire is equal to T=1.5N. The ring is placed in a magnetic field, which is perpendicular to the plane of th ering so that the forces tend to break the ring. Find B ( in T) at which the ring is broken.

Options

A. 0.5 T
B. 1 T
C. 2 T
D. 2.5 T

Explanation

To solve the problem, we need to find the magnetic field \( B \) at which the ring breaks due to the tensile strength of the wire. Here’s a step-by-step solution:

### Step 1: Understand the Forces Acting on the Ring
When a current \( I \) flows through the ring in a magnetic field \( B \), a magnetic force acts on the wire. This force tends to stretch the wire, and we need to find the magnetic field strength at which this force equals the tensile strength \( T \) of the wire.

### Step 2: Calculate the Magnetic Force on the Ring
The magnetic force \( F \) acting on a segment of the ring can be expressed as:
\[ F = B \cdot I \cdot L \]
where:
- \( B \) is the magnetic field strength,
- \( I \) is the current flowing through the wire,
- \( L \) is the length of the wire segment in the magnetic field.

For a ring, the length \( L \) is the circumference of the ring:
\[ L = 2\pi r_0 \]
where \( r_0 \) is the radius of the ring.

### Step 3: Substitute the Length of the Ring
Substituting the expression for \( L \) into the force equation, we have:
\[ F = B \cdot I \cdot (2\pi r_0) \]

### Step 4: Set the Magnetic Force Equal to the Tensile Strength
The ring will break when the magnetic force equals the tensile strength \( T \):
\[ B \cdot I \cdot (2\pi r_0) = T \]

### Step 5: Solve for the Magnetic Field \( B \)
Rearranging the equation to solve for \( B \):
\[ B = \frac{T}{I \cdot (2\pi r_0)} \]

### Step 6: Substitute the Given Values
Now, we can substitute the known values:
- \( T = 1.5 \, \text{N} \)
- \( I = 10 \, \text{A} \)
- \( r_0 = 15 \, \text{cm} = 0.15 \, \text{m} \)

Substituting these values into the equation:
\[ B = \frac{1.5}{10 \cdot (2\pi \cdot 0.15)} \]

### Step 7: Calculate the Value
Calculating the denominator:
\[ 2\pi \cdot 0.15 \approx 0.942 \, \text{m} \]
Now substituting this back into the equation for \( B \):
\[ B = \frac{1.5}{10 \cdot 0.942} \approx \frac{1.5}{9.42} \approx 0.159 \, \text{T} \]

### Step 8: Final Calculation
Calculating the final value gives:
\[ B \approx 0.159 \, \text{T} \]

### Conclusion
The magnetic field strength at which the ring will break is approximately \( 0.159 \, \text{T} \).

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